Polar Form Calculator
Omni's polar form calculator can help whenever you want to convert a complex number from rectangular to polar form. Scroll down to find a short article explaining what the different forms of a complex number are all about and what the polar form conversion formulas look like. Believe it or not, converting to polar form can be fun!
What are the forms of a complex number?
The two main forms of a complex number are polar and rectangular. Usually, you first encounter the rectangular form
z = a + bi, where
b are the real part and imaginary part of
z, respectively. They are both real numbers!
The polar form describes
r × exp(φi), where:
- The magnitude
ris the distance from the origin
- The argument
φis the angle between the real axis (x-axis) and the radius connecting the origin
Look at the picture to better understand what the polar form is:
How do I convert from rectangular to polar form?
To convert a complex number from the rectangular
a + bi to polar form
z, we use the formulas:
r = √(a² + b²)
φ = atan2(b, a).
a > 0, then
atan2(b, a) = arctan(b / a). However, if
a < 0, then
atan(b /a) does not give us the correct angle - it must be corrected by
±π to send us into the correct quadrant of the plane. Formally
atan2(b, a) is defined as:
atan(b / a)if
a > 0;
atan(b / a) + πif
a < 0 ≤ b;
atan(b / a) - πif
a,b < 0;
a = 0 < b;
b < 0 = a; and
- remains undefined if
x = y = 0.
These are exactly the formulas behind Omni's polar form converter.
How to use this polar form calculator?
To use this polar form converter, take a complex number in the rectangular form
a + bi and input
b into the respective fields of our tool. The two ingredients of the polar form will appear immediately in their fields: the magnitude
r and phase (argument)
φ. Take them and write down the polar form
r × exp(iφ) of your number.
This can't get any easier!
Omni tools for complex numbers
Complex numbers can find you in sooo many areas of science! Omni is well aware of that - we have built a whole collection of tools addressing different problems related to complex numbers. Make sure to check out at least a few of the following tools:
- Complex number calculator;
- a+bi form calculator;
- Multiply complex numbers calculator;
- Divide complex numbers calculator;
- Imaginary number calculator;
- Complex number to polar form calculator;
- Complex number to trigonometric form calculator;
- Complex number to rectangular form calculator; and
- i calculator.
How do I convert from trigonometric to polar form?
To convert a complex number from trigonometric to polar form, you need to:
- Make sure your number is written as
z = r × cos(φ) + r × i × sin(φ)].
- Extract the magnitude
rand the argument
- Write down your number as
r × exp(φi).
- If in doubt, use an online polar form calculator.
What is the polar form of -1?
The answer is
exp(πi). To derive this answer, observe that the modulus (magnitude) of
-1 is equal to
1. Next, we must determine
φ such that
cos(φ) = -1 and
sin(φ) = 0. We easily verify that
φ = π satisfies these conditions. Hence, the whole number reads
1 × exp(πi), as claimed.